Candidate Models

While summed ordinal scales are often treated as though they are scale data, analysed using models with a Gaussian likelihood, this assumes that the data are continuous and unbounded, with equal distance between measurements. This can result in model-based predictions that are impossible under the scale in question. Instead, analysing the data using an ordinal model allows the restrictions of the ordinal scale to be taken into account. While this is often done with trial-level data, we modelled the sum scores of the ordinal scale using an ordinal model under the assumption that many of the restrictions for modelling ordinal data still apply while using sum scores to allow for easy interpretation of coefficients from each scale for diagnostic value. Below, we present posterior predictive checks for several potential candidate models to show that the ordinal model best captures the data generation process. This is carried out for the depression data only using the same predictors and outcomes as used in the main models reported in the paper. The considered models are:

The ordinal model (cumulative likelihood with a logit link): Data arise from categorisation of a continuous latent variable.
Beta binomial likelihood: Data are non-negative integers with a varying probability of success from n trials arising from a Beta distribution.
Negative binomial likelihood: Data are non-negative integers representing the number of failures from a sequence of i.i.d. Bernoulli trials.
The traditional” Gaussian likelihood: Data are unbounded and continuous.
Poisson likelihood: Data are non-negative integers representing the number of events occurring in a fixed interval of time or space.

Conceptually, the ordinal and Beta binomial models are most aligned with the data generation process. Below, dark lines represent the observed density of the data while light lines represent model-based predictions. A well behaving model should be able to generate the observed data such that the shape of the two line types should be similar.

It is apparent that the ordinal model best captures the data generation process and thus is an appropriate model for analysing these data. Note that predictions for the ordinal model represent the thresholds for each ordinal category, which is roughly half of the actual sum score.

Full Parameter Estimates

Full fixed effect parameter estimates are included for all models below. Note that these are presented on the log scale. In the paper only the beta terms are included along with Bayes factors evaluating evidence in support of the point-null hypothesis relative to the alternative hypothesis (i.e. of a non-zero effect) using the Savage-Dickey density ratio, but not with intercepts or thresholds.

Cumulative models assume that responses come from categoristion of a latent continuous variable. Thus, for the cumulative models there is not a single intercept but instead the model output reports several “intercepts” which are thresholds that partition the latent continuous variable into response categories. Holding all other parameters constant, these give the cumulative log-odds of responses being within a given threshold. (To transform these into probabilities, the inverse logit should be applied to any given threshold.) As with intercepts in typical regression modelling, these are often not the focus of the research question. Thus, intercepts and thresholds are omitted from the main text but are included in full here for completeness.

Crucially, the thresholds in the cumulative models reported here are dictated by the unique scores on the outcome. In the depression, anxiety, and stress models each “intercept” typically represents scores from 0 to 42 in increments of 2 as the scale requires individual scores to be summed and multiplied by 2. However, in the short loneliness scale used in study 1, the thresholds represent scores from 3 to 9 in increments of 1. For the longer loneliness scale used in studies 2 and 3, the thresholds represent scores from 0 to 11 in increments of 1. In cases where the thresholds do not match the number of unique increments on the scale this is due to some unique values not occurring in the data set.

As the tables are explained fully in text, no further interpretation is given for each table.

Study 1

Main Model

term	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Depression
Intercept[1]	-4.446	0.364	-5.173	-3.743	1	2708	4429
Intercept[2]	-3.193	0.305	-3.811	-2.607	1	2786	4933
Intercept[3]	-2.503	0.280	-3.081	-1.976	1	2785	4774
Intercept[4]	-2.010	0.268	-2.552	-1.493	1	2728	4963
Intercept[5]	-1.284	0.253	-1.788	-0.799	1	2682	4818
Intercept[6]	-0.583	0.244	-1.061	-0.109	1	2829	5070
Intercept[7]	-0.117	0.243	-0.584	0.362	1	2968	5409
Intercept[8]	0.395	0.243	-0.071	0.880	1	2960	5350
Intercept[9]	0.915	0.246	0.435	1.402	1	3048	5314
Intercept[10]	1.372	0.254	0.882	1.878	1	3154	4487
Intercept[11]	1.744	0.259	1.254	2.262	1	3179	5043
Intercept[12]	2.193	0.272	1.679	2.747	1	3253	5292
Intercept[13]	2.519	0.280	1.996	3.093	1	3240	5276
Intercept[14]	2.756	0.288	2.206	3.339	1	3265	5019
Intercept[15]	3.202	0.305	2.619	3.829	1	3412	5071
Intercept[16]	3.926	0.332	3.293	4.600	1	3625	5399
Intercept[17]	4.304	0.350	3.642	5.009	1	3856	5793
Intercept[18]	4.824	0.375	4.095	5.572	1	3939	5591
Intercept[19]	5.376	0.414	4.583	6.207	1	4131	5784
Intercept[20]	5.972	0.458	5.096	6.898	1	4401	6349
Intercept[21]	6.867	0.546	5.838	7.998	1	5002	6385
Lockdown Period	0.451	0.099	0.257	0.645	1	7956	6747
Total Hours	0.360	0.173	0.021	0.700	1	4170	5539
Lockdown Period by Hours	-0.072	0.106	-0.275	0.135	1	10559	7063
Anxiety
Intercept[1]	-3.560	0.353	-4.273	-2.905	1	2116	3968
Intercept[2]	-1.703	0.284	-2.277	-1.171	1	2233	4186
Intercept[3]	-0.687	0.263	-1.204	-0.181	1	2483	4792
Intercept[4]	-0.051	0.260	-0.561	0.463	1	2453	4517
Intercept[5]	0.871	0.264	0.351	1.389	1	2604	4148
Intercept[6]	1.490	0.274	0.973	2.038	1	2676	4885
Intercept[7]	2.152	0.289	1.604	2.731	1	2755	4799
Intercept[8]	2.989	0.318	2.390	3.644	1	3003	4837
Intercept[9]	3.752	0.350	3.096	4.461	1	3230	5122
Intercept[10]	4.500	0.387	3.761	5.276	1	3600	5493
Intercept[11]	5.393	0.436	4.564	6.285	1	4008	5500
Intercept[12]	5.901	0.465	5.042	6.851	1	4271	5758
Intercept[13]	6.290	0.494	5.363	7.305	1	4476	5427
Intercept[14]	6.906	0.554	5.873	8.036	1	4912	5937
Intercept[15]	7.647	0.635	6.481	8.924	1	5464	6358
Intercept[16]	8.560	0.780	7.153	10.206	1	6382	5739
Intercept[17]	9.278	0.942	7.628	11.318	1	6879	5538
Intercept[18]	10.622	1.355	8.362	13.693	1	9188	6048
Lockdown Period	0.191	0.099	-0.002	0.387	1	8096	6130
Total Hours	0.333	0.174	0.001	0.683	1	3937	5259
Lockdown Period by Hours	-0.081	0.107	-0.295	0.131	1	10132	6493
Stress
Intercept[1]	-4.847	0.389	-5.655	-4.111	1	2445	3864
Intercept[2]	-3.920	0.336	-4.621	-3.284	1	2459	4118
Intercept[3]	-3.132	0.295	-3.739	-2.574	1	2422	3743
Intercept[4]	-2.383	0.265	-2.917	-1.872	1	2599	4366
Intercept[5]	-1.686	0.245	-2.180	-1.214	1	2670	4433
Intercept[6]	-0.982	0.233	-1.450	-0.528	1	2786	4657
Intercept[7]	-0.228	0.227	-0.679	0.222	1	2885	4379
Intercept[8]	0.433	0.227	-0.004	0.877	1	2989	4804
Intercept[9]	0.955	0.231	0.513	1.423	1	3110	4610
Intercept[10]	1.551	0.238	1.089	2.042	1	3119	4661
Intercept[11]	2.149	0.253	1.674	2.672	1	3249	4851
Intercept[12]	2.577	0.264	2.082	3.125	1	3281	5248
Intercept[13]	3.174	0.285	2.640	3.762	1	3468	5238
Intercept[14]	3.697	0.305	3.125	4.333	1	3529	5567
Intercept[15]	4.247	0.333	3.622	4.913	1	3656	4813
Intercept[16]	4.699	0.357	4.028	5.419	1	3719	5324
Intercept[17]	4.915	0.369	4.213	5.670	1	3694	5461
Intercept[18]	5.947	0.449	5.094	6.836	1	4257	5760
Intercept[19]	6.840	0.555	5.823	7.983	1	5261	5385
Intercept[20]	7.734	0.709	6.458	9.272	1	6499	6294
Intercept[21]	9.302	1.179	7.397	11.992	1	9224	6164
Lockdown Period	0.388	0.097	0.197	0.576	1	7318	6308
Total Hours	0.116	0.157	-0.195	0.423	1	3521	5316
Lockdown Period by Hours	0.053	0.104	-0.151	0.258	1	8079	5886
Loneliness
Intercept[1]	-2.531	0.288	-3.115	-1.998	1	2953	4888
Intercept[2]	-1.133	0.247	-1.631	-0.666	1	2962	4979
Intercept[3]	0.126	0.240	-0.345	0.597	1	3129	5454
Intercept[4]	1.886	0.270	1.372	2.428	1	3303	4045
Intercept[5]	3.218	0.316	2.628	3.867	1	3614	5057
Intercept[6]	4.443	0.384	3.721	5.230	1	3932	4907
Lockdown Period	0.620	0.108	0.410	0.833	1	8316	5558
Total Hours	0.013	0.171	-0.328	0.341	1	5076	6245
Lockdown Period by Hours	0.073	0.113	-0.146	0.299	1	10314	6402

Moderation Model

term	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Depression
Intercept[1]	1.169	0.448	0.277	2.056	1	10173	6821
Intercept[2]	1.862	0.437	1.016	2.743	1	9830	7187
Intercept[3]	2.396	0.436	1.549	3.278	1	9306	7066
Intercept[4]	2.732	0.441	1.872	3.619	1	9352	7190
Intercept[5]	3.037	0.448	2.166	3.941	1	9303	7335
Intercept[6]	3.504	0.462	2.615	4.425	1	9249	7092
Intercept[7]	3.799	0.470	2.891	4.737	1	9190	6928
Intercept[8]	4.155	0.480	3.233	5.115	1	9277	6991
Intercept[9]	4.440	0.490	3.495	5.425	1	9380	6936
Intercept[10]	4.686	0.499	3.726	5.685	1	9347	6865
Intercept[11]	4.913	0.508	3.938	5.928	1	9325	6815
Intercept[12]	5.194	0.518	4.212	6.230	1	9394	6826
Intercept[13]	5.428	0.524	4.442	6.485	1	9528	7160
Intercept[14]	5.534	0.526	4.535	6.604	1	9523	6992
Intercept[15]	5.911	0.540	4.895	6.997	1	9460	6870
Intercept[16]	6.358	0.555	5.299	7.472	1	9644	6438
Intercept[17]	6.647	0.566	5.557	7.775	1	9858	6458
Intercept[18]	6.929	0.576	5.821	8.099	1	9972	6734
Intercept[19]	7.387	0.595	6.241	8.593	1	10210	6061
Intercept[20]	7.599	0.605	6.455	8.810	1	10389	6709
Intercept[21]	8.132	0.642	6.902	9.408	1	10616	6687
Difference in Hours Played	0.013	0.022	-0.028	0.057	1	10203	5551
Loneliness During Lockdown	0.699	0.079	0.550	0.855	1	9632	6605
Hours by Loneliness	-0.001	0.004	-0.008	0.006	1	10067	5659
Anxiety
Intercept[1]	0.660	0.432	-0.177	1.520	1	9896	7050
Intercept[2]	1.498	0.437	0.657	2.365	1	9594	6663
Intercept[3]	2.100	0.445	1.257	2.980	1	9397	6743
Intercept[4]	2.369	0.453	1.513	3.281	1	9295	7151
Intercept[5]	2.759	0.465	1.883	3.696	1	9092	7253
Intercept[6]	2.944	0.470	2.056	3.891	1	9030	7200
Intercept[7]	3.286	0.481	2.361	4.257	1	9100	7151
Intercept[8]	3.761	0.494	2.809	4.760	1	9122	7015
Intercept[9]	4.197	0.507	3.216	5.214	1	8848	6441
Intercept[10]	4.556	0.520	3.548	5.609	1	8996	6610
Intercept[11]	5.139	0.550	4.077	6.259	1	8796	6870
Intercept[12]	5.567	0.572	4.478	6.728	1	9499	6926
Intercept[13]	5.841	0.593	4.714	7.038	1	10120	6930
Intercept[14]	6.473	0.650	5.246	7.787	1	10264	6811
Intercept[15]	7.083	0.736	5.714	8.577	1	10651	6968
Intercept[16]	7.722	0.874	6.133	9.590	1	10737	6622
Intercept[17]	8.867	1.225	6.832	11.660	1	12393	6346
Difference in Hours Played	0.030	0.022	-0.012	0.073	1	8528	6298
Loneliness During Lockdown	0.398	0.076	0.250	0.551	1	9196	6575
Hours by Loneliness	-0.003	0.004	-0.010	0.004	1	8450	6130
Stress
Intercept[1]	0.326	0.449	-0.562	1.196	1	10986	6654
Intercept[2]	0.841	0.435	-0.019	1.694	1	11227	6696
Intercept[3]	1.195	0.430	0.361	2.032	1	11012	6872
Intercept[4]	1.618	0.432	0.773	2.467	1	10573	6620
Intercept[5]	2.010	0.438	1.162	2.874	1	10161	6885
Intercept[6]	2.404	0.446	1.537	3.288	1	9828	6715
Intercept[7]	2.761	0.454	1.878	3.662	1	9734	6706
Intercept[8]	3.183	0.464	2.292	4.102	1	9830	6534
Intercept[9]	3.436	0.472	2.520	4.358	1	9739	6527
Intercept[10]	3.853	0.484	2.902	4.807	1	9712	6620
Intercept[11]	4.203	0.494	3.242	5.193	1	9844	6827
Intercept[12]	4.428	0.500	3.451	5.418	1	9879	6553
Intercept[13]	4.938	0.519	3.928	5.968	1	10134	6701
Intercept[14]	5.272	0.528	4.254	6.313	1	10246	6567
Intercept[15]	5.589	0.537	4.560	6.657	1	10376	6467
Intercept[16]	5.773	0.545	4.718	6.854	1	10411	6242
Intercept[17]	5.976	0.552	4.919	7.061	1	10385	6607
Intercept[18]	6.560	0.583	5.433	7.726	1	10993	6510
Intercept[19]	7.312	0.647	6.088	8.602	1	11868	6512
Intercept[20]	8.123	0.778	6.688	9.743	1	13028	6317
Intercept[21]	9.574	1.183	7.615	12.282	1	13619	6416
Difference in Hours Played	0.025	0.022	-0.017	0.068	1	9362	6687
Loneliness During Lockdown	0.531	0.076	0.383	0.682	1	9656	6624
Hours by Loneliness	-0.002	0.004	-0.010	0.005	1	9492	5892

Study 2

Main Model

term	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Depression
Intercept[1]	-8.129	1.068	-10.408	-6.226	1	2577	2973
Intercept[2]	-5.793	0.802	-7.474	-4.334	1	2310	3417
Intercept[3]	-4.232	0.675	-5.668	-3.023	1	2282	3782
Intercept[4]	-3.061	0.584	-4.265	-2.001	1	2301	3463
Intercept[5]	-2.132	0.528	-3.242	-1.163	1	2381	3789
Intercept[6]	-1.193	0.487	-2.196	-0.297	1	2485	4045
Intercept[7]	-0.282	0.463	-1.201	0.625	1	2687	3790
Intercept[8]	0.341	0.456	-0.562	1.227	1	2930	4530
Intercept[9]	0.869	0.454	-0.027	1.765	1	3018	4901
Intercept[10]	1.208	0.459	0.319	2.117	1	2996	4677
Intercept[11]	1.564	0.469	0.670	2.497	1	3031	4484
Intercept[12]	1.714	0.474	0.813	2.669	1	3051	4491
Intercept[13]	2.529	0.512	1.573	3.562	1	3242	4971
Intercept[14]	3.320	0.566	2.271	4.490	1	3343	5008
Intercept[15]	4.019	0.612	2.880	5.288	1	3243	4932
Intercept[16]	4.657	0.663	3.443	6.036	1	3516	5085
Intercept[17]	5.212	0.696	3.927	6.654	1	3699	5000
Intercept[18]	5.719	0.738	4.346	7.260	1	3790	5367
Intercept[19]	6.348	0.792	4.871	7.992	1	3672	5842
Intercept[20]	6.737	0.827	5.194	8.463	1	3882	5903
Intercept[21]	8.326	1.069	6.417	10.610	1	4507	6010
Lockdown Period	0.175	0.149	-0.115	0.464	1	13193	5659
Total Hours	0.803	0.265	0.294	1.334	1	5216	5464
Lockdown Period by Hours	0.004	0.167	-0.324	0.331	1	10430	5660
Anxiety
Intercept[1]	-4.919	0.732	-6.441	-3.580	1	1309	2574
Intercept[2]	-2.981	0.596	-4.206	-1.887	1	1304	2416
Intercept[3]	-1.384	0.514	-2.433	-0.447	1	1335	2467
Intercept[4]	-0.075	0.476	-1.044	0.832	1	1355	2712
Intercept[5]	0.731	0.472	-0.193	1.663	1	1489	2949
Intercept[6]	1.752	0.497	0.815	2.737	1	1829	3527
Intercept[7]	2.119	0.508	1.162	3.159	1	1960	3663
Intercept[8]	2.922	0.545	1.904	4.067	1	2374	4806
Intercept[9]	3.470	0.580	2.406	4.666	1	2717	4695
Intercept[10]	3.911	0.613	2.795	5.213	1	2957	5078
Intercept[11]	4.669	0.687	3.399	6.106	1	3316	4680
Intercept[12]	5.757	0.780	4.322	7.396	1	3593	4535
Intercept[13]	6.071	0.805	4.622	7.761	1	3579	4912
Intercept[14]	7.095	0.916	5.420	8.998	1	3952	5212
Intercept[15]	7.691	0.997	5.879	9.770	1	4452	5935
Intercept[16]	9.344	1.382	6.971	12.310	1	5682	5964
Lockdown Period	0.074	0.149	-0.219	0.366	1	9433	6185
Total Hours	0.600	0.258	0.102	1.104	1	2958	4535
Lockdown Period by Hours	0.175	0.165	-0.150	0.501	1	7451	6102
Stress
Intercept[1]	-7.554	1.016	-9.681	-5.729	1	2328	3654
Intercept[2]	-6.072	0.832	-7.770	-4.539	1	2303	3437
Intercept[3]	-4.940	0.727	-6.459	-3.609	1	2197	3696
Intercept[4]	-3.712	0.637	-5.071	-2.546	1	2121	3600
Intercept[5]	-2.555	0.566	-3.741	-1.509	1	2052	3292
Intercept[6]	-1.770	0.522	-2.844	-0.798	1	2080	3638
Intercept[7]	-0.599	0.478	-1.556	0.311	1	2063	4014
Intercept[8]	-0.251	0.474	-1.205	0.654	1	2112	3662
Intercept[9]	0.485	0.473	-0.442	1.405	1	2189	3650
Intercept[10]	1.164	0.484	0.235	2.129	1	2221	3837
Intercept[11]	2.091	0.513	1.118	3.157	1	2236	3979
Intercept[12]	2.738	0.534	1.727	3.856	1	2344	3843
Intercept[13]	3.071	0.546	2.053	4.210	1	2446	3651
Intercept[14]	3.796	0.578	2.724	4.989	1	2594	4246
Intercept[15]	4.805	0.646	3.626	6.148	1	2835	4838
Intercept[16]	5.271	0.680	4.040	6.697	1	3066	4797
Intercept[17]	6.233	0.777	4.843	7.890	1	3614	5721
Intercept[18]	6.700	0.847	5.189	8.492	1	3854	5171
Intercept[19]	7.310	0.945	5.622	9.388	1	4172	5810
Intercept[20]	9.143	1.449	6.699	12.421	1	4399	5888
Lockdown Period	-0.073	0.142	-0.345	0.212	1	10153	5768
Total Hours	0.628	0.261	0.120	1.151	1	4154	5345
Lockdown Period by Hours	0.254	0.166	-0.068	0.581	1	8075	6024
Loneliness
Intercept[1]	-9.306	1.506	-12.656	-6.720	1	2008	3113
Intercept[2]	-6.705	1.022	-8.932	-4.906	1	2207	3303
Intercept[3]	-4.850	0.780	-6.500	-3.489	1	2214	3475
Intercept[4]	-2.950	0.599	-4.229	-1.867	1	2338	3776
Intercept[5]	-2.190	0.542	-3.332	-1.179	1	2363	3763
Intercept[6]	-1.181	0.490	-2.173	-0.250	1	2342	3954
Intercept[7]	-0.492	0.474	-1.409	0.434	1	2266	3716
Intercept[8]	0.851	0.492	-0.056	1.843	1	2124	3514
Intercept[9]	1.784	0.530	0.811	2.881	1	2061	3407
Intercept[10]	2.995	0.594	1.920	4.225	1	2096	3408
Intercept[11]	5.055	0.730	3.729	6.596	1	2088	4084
Lockdown Period	0.139	0.151	-0.159	0.437	1	10423	5682
Total Hours	0.212	0.259	-0.298	0.718	1	3971	5380
Lockdown Period by Hours	0.008	0.171	-0.330	0.341	1	8264	6196

Moderation Model

term	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Depression
Intercept[1]	-1.244	0.659	-2.596	-0.019	1	10853	5950
Intercept[2]	-0.069	0.533	-1.107	0.981	1	13733	6713
Intercept[3]	0.880	0.514	-0.111	1.906	1	12442	7269
Intercept[4]	1.190	0.517	0.189	2.236	1	11996	6713
Intercept[5]	1.673	0.531	0.655	2.746	1	11571	6694
Intercept[6]	2.050	0.546	1.001	3.142	1	11302	6751
Intercept[7]	2.340	0.558	1.282	3.477	1	11073	6564
Intercept[8]	2.745	0.573	1.663	3.898	1	10780	6730
Intercept[9]	3.106	0.589	1.988	4.278	1	10749	6856
Intercept[10]	3.293	0.596	2.150	4.486	1	10698	6430
Intercept[11]	3.487	0.605	2.329	4.699	1	10824	6582
Intercept[12]	3.689	0.614	2.523	4.925	1	10863	6528
Intercept[13]	4.051	0.637	2.844	5.320	1	10790	6298
Intercept[14]	4.373	0.657	3.123	5.676	1	10813	6327
Intercept[15]	4.546	0.671	3.264	5.878	1	10826	6406
Intercept[16]	4.817	0.690	3.521	6.191	1	11065	6552
Intercept[17]	5.226	0.713	3.874	6.659	1	11219	6580
Intercept[18]	5.750	0.755	4.319	7.279	1	11672	6783
Intercept[19]	6.090	0.782	4.607	7.672	1	11577	6981
Intercept[20]	7.270	0.988	5.495	9.400	1	12680	6389
Difference in Hours Played	-0.008	0.027	-0.061	0.045	1	10144	6248
Loneliness During Lockdown	0.350	0.070	0.219	0.488	1	12083	6958
Hours by Loneliness	0.003	0.003	-0.004	0.009	1	10310	6272
Anxiety
Intercept[1]	-0.313	0.494	-1.308	0.645	1	7965	6161
Intercept[2]	0.551	0.479	-0.367	1.477	1	9102	6689
Intercept[3]	1.057	0.486	0.123	2.012	1	8418	6638
Intercept[4]	1.653	0.504	0.683	2.655	1	8014	6817
Intercept[5]	2.003	0.520	1.010	3.033	1	7867	6878
Intercept[6]	2.305	0.532	1.271	3.367	1	7463	6810
Intercept[7]	2.550	0.542	1.495	3.629	1	7495	6791
Intercept[8]	2.885	0.558	1.802	4.007	1	7534	6502
Intercept[9]	3.108	0.571	2.007	4.271	1	7363	6496
Intercept[10]	3.440	0.590	2.315	4.624	1	7586	6775
Intercept[11]	3.726	0.609	2.558	4.963	1	7788	6628
Intercept[12]	4.061	0.634	2.870	5.353	1	8519	6741
Intercept[13]	4.326	0.659	3.076	5.632	1	8676	7136
Intercept[14]	5.320	0.793	3.857	6.994	1	8939	6931
Intercept[15]	6.783	1.187	4.762	9.435	1	10142	6359
Difference in Hours Played	0.017	0.026	-0.033	0.070	1	6094	4348
Loneliness During Lockdown	0.220	0.064	0.095	0.349	1	7270	5731
Hours by Loneliness	-0.001	0.003	-0.007	0.005	1	6385	4947
Stress
Intercept[1]	-1.867	0.635	-3.168	-0.670	1	9482	6356
Intercept[2]	-1.211	0.570	-2.341	-0.093	1	10822	6706
Intercept[3]	-0.680	0.541	-1.735	0.385	1	10998	6310
Intercept[4]	-0.012	0.520	-1.017	1.017	1	11805	6467
Intercept[5]	0.535	0.514	-0.456	1.560	1	11576	6547
Intercept[6]	0.815	0.514	-0.160	1.838	1	11740	6759
Intercept[7]	1.227	0.514	0.253	2.266	1	11677	6877
Intercept[8]	1.328	0.516	0.342	2.355	1	11563	6483
Intercept[9]	1.688	0.528	0.700	2.754	1	11382	6595
Intercept[10]	2.012	0.544	0.992	3.094	1	11245	6506
Intercept[11]	2.127	0.548	1.093	3.213	1	11215	6400
Intercept[12]	2.478	0.565	1.412	3.601	1	10880	6512
Intercept[13]	2.731	0.575	1.652	3.868	1	11107	6479
Intercept[14]	3.249	0.599	2.120	4.449	1	11070	6837
Intercept[15]	3.835	0.632	2.656	5.102	1	11353	6783
Intercept[16]	4.232	0.671	2.985	5.597	1	11322	6331
Intercept[17]	4.764	0.736	3.398	6.284	1	11337	6748
Intercept[18]	5.262	0.813	3.777	6.937	1	11324	6709
Intercept[19]	6.735	1.201	4.694	9.473	1	11764	6318
Difference in Hours Played	0.050	0.026	0.000	0.101	1	9122	6275
Loneliness During Lockdown	0.207	0.067	0.079	0.337	1	11729	6400
Hours by Loneliness	-0.004	0.003	-0.011	0.002	1	9086	6622

Study 3

Main Model

term	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Depression
Intercept[1]	-6.862	1.013	-9.014	-5.016	1	2019	3197
Intercept[2]	-4.816	0.774	-6.436	-3.431	1	1948	3181
Intercept[3]	-3.863	0.677	-5.283	-2.633	1	2001	3409
Intercept[4]	-3.062	0.608	-4.315	-1.964	1	1973	3372
Intercept[5]	-2.214	0.550	-3.354	-1.230	1	1946	3401
Intercept[6]	-1.404	0.508	-2.444	-0.470	1	1919	3687
Intercept[7]	-1.096	0.500	-2.122	-0.159	1	1961	3589
Intercept[8]	-0.133	0.483	-1.095	0.797	1	2040	3665
Intercept[9]	0.217	0.483	-0.739	1.148	1	2046	3867
Intercept[10]	0.566	0.485	-0.390	1.529	1	2104	3603
Intercept[11]	1.194	0.498	0.226	2.189	1	2129	3369
Intercept[12]	1.815	0.520	0.833	2.885	1	2369	4035
Intercept[13]	2.540	0.559	1.484	3.688	1	2478	3658
Intercept[14]	3.009	0.593	1.920	4.244	1	2510	3996
Intercept[15]	3.534	0.629	2.376	4.837	1	2720	4601
Intercept[16]	4.326	0.695	3.056	5.770	1	2787	4545
Intercept[17]	5.002	0.764	3.617	6.578	1	2982	4866
Intercept[18]	5.651	0.851	4.142	7.438	1	3227	5240
Intercept[19]	6.224	0.932	4.590	8.190	1	3435	5639
Intercept[20]	7.455	1.151	5.444	9.864	1	4266	5797
Lockdown Period	0.091	0.173	-0.244	0.429	1	7024	5940
Total Hours	0.032	0.365	-0.692	0.751	1	2365	3660
Lockdown Period by Hours	0.198	0.189	-0.176	0.573	1	7716	6175
Anxiety
Intercept[1]	-5.224	1.048	-7.506	-3.387	1	1139	2856
Intercept[2]	-3.532	0.882	-5.441	-1.981	1	1145	2812
Intercept[3]	-2.209	0.753	-3.794	-0.849	1	1147	2476
Intercept[4]	-1.215	0.696	-2.664	0.073	1	1183	2547
Intercept[5]	-0.061	0.657	-1.378	1.184	1	1289	2513
Intercept[6]	0.763	0.649	-0.532	2.025	1	1364	2887
Intercept[7]	1.256	0.653	-0.040	2.540	1	1450	2944
Intercept[8]	2.037	0.668	0.770	3.370	1	1611	3211
Intercept[9]	2.730	0.701	1.367	4.163	1	1791	3259
Intercept[10]	3.418	0.755	2.013	4.970	1	2079	4012
Intercept[11]	3.723	0.778	2.275	5.334	1	2180	3962
Intercept[12]	4.538	0.852	2.962	6.316	1	2424	3695
Intercept[13]	5.272	0.917	3.605	7.243	1	2702	4289
Intercept[14]	6.103	0.996	4.273	8.183	1	2995	4193
Intercept[15]	6.590	1.046	4.703	8.781	1	3217	4602
Intercept[16]	9.072	1.604	6.391	12.609	1	4518	5648
Intercept[17]	11.400	2.333	7.620	16.791	1	4801	4964
Lockdown Period	0.120	0.183	-0.239	0.472	1	9160	6052
Total Hours	-0.222	0.386	-0.995	0.521	1	2344	3993
Lockdown Period by Hours	-0.079	0.202	-0.476	0.318	1	8277	6488
Stress
Intercept[1]	-7.315	1.148	-9.772	-5.280	1	2703	3662
Intercept[2]	-6.145	0.980	-8.230	-4.391	1	2683	4218
Intercept[3]	-5.381	0.899	-7.287	-3.779	1	2705	4033
Intercept[4]	-4.321	0.796	-5.994	-2.897	1	2666	4214
Intercept[5]	-3.909	0.757	-5.501	-2.544	1	2591	3903
Intercept[6]	-2.572	0.644	-3.908	-1.401	1	2535	3861
Intercept[7]	-1.381	0.576	-2.546	-0.304	1	2502	4190
Intercept[8]	-0.805	0.559	-1.951	0.263	1	2549	4366
Intercept[9]	-0.057	0.552	-1.151	1.032	1	2653	4503
Intercept[10]	0.323	0.550	-0.769	1.418	1	2613	4235
Intercept[11]	0.905	0.559	-0.194	2.040	1	2709	4461
Intercept[12]	1.873	0.590	0.762	3.062	1	2916	4853
Intercept[13]	2.333	0.608	1.178	3.543	1	2998	4974
Intercept[14]	3.048	0.650	1.868	4.416	1	3096	5234
Intercept[15]	3.543	0.692	2.267	4.993	1	3271	5543
Intercept[16]	5.214	0.890	3.589	7.094	1	3713	5037
Intercept[17]	6.157	1.015	4.324	8.254	1	3878	5100
Intercept[18]	7.147	1.185	5.064	9.684	1	3889	4905
Intercept[19]	8.425	1.432	5.911	11.493	1	3844	4634
Intercept[20]	10.027	1.696	7.013	13.708	1	4369	5460
Lockdown Period	-0.002	0.175	-0.348	0.346	1	11602	5922
Total Hours	-0.128	0.353	-0.822	0.574	1	3946	5263
Lockdown Period by Hours	0.237	0.189	-0.138	0.613	1	9330	6316
Loneliness
Intercept[1]	-8.966	1.753	-12.941	-6.081	1	1977	3121
Intercept[2]	-7.584	1.390	-10.733	-5.204	1	1995	3694
Intercept[3]	-5.730	1.008	-7.900	-3.966	1	1786	3608
Intercept[4]	-4.065	0.768	-5.691	-2.714	1	1769	3810
Intercept[5]	-2.752	0.639	-4.090	-1.618	1	1619	3201
Intercept[6]	-1.673	0.568	-2.865	-0.620	1	1624	3194
Intercept[7]	0.037	0.509	-0.950	1.045	1	1903	3597
Intercept[8]	0.881	0.528	-0.102	1.979	1	1918	3521
Intercept[9]	1.629	0.564	0.590	2.797	1	2060	3891
Intercept[10]	2.927	0.652	1.738	4.281	1	2186	3992
Intercept[11]	5.245	0.858	3.702	7.082	1	2598	4338
Lockdown Period	0.153	0.181	-0.200	0.510	1	7045	5374
Total Hours	0.253	0.342	-0.397	0.949	1	2218	3794
Lockdown Period by Hours	-0.090	0.184	-0.458	0.268	1	6485	5733

Moderation Model

term	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
Depression
Intercept[1]	-1.364	0.973	-3.397	0.444	1	6461	5301
Intercept[2]	0.232	0.757	-1.257	1.729	1	9280	6869
Intercept[3]	0.637	0.738	-0.799	2.095	1	9055	6566
Intercept[4]	1.094	0.726	-0.314	2.545	1	8996	6674
Intercept[5]	1.749	0.736	0.360	3.242	1	8759	6940
Intercept[6]	2.001	0.750	0.585	3.550	1	8695	6876
Intercept[7]	2.262	0.765	0.829	3.824	1	8609	7110
Intercept[8]	2.811	0.797	1.296	4.454	1	8394	6527
Intercept[9]	3.023	0.806	1.496	4.660	1	8441	6499
Intercept[10]	3.234	0.814	1.678	4.884	1	8278	6415
Intercept[11]	3.681	0.836	2.085	5.363	1	8170	6550
Intercept[12]	4.004	0.857	2.365	5.730	1	8059	6643
Intercept[13]	4.465	0.886	2.784	6.273	1	7907	6611
Intercept[14]	4.679	0.895	2.976	6.506	1	7854	6385
Intercept[15]	5.036	0.917	3.314	6.882	1	7870	6497
Intercept[16]	5.305	0.933	3.539	7.206	1	7861	6700
Intercept[17]	5.784	0.977	3.946	7.775	1	8092	6331
Intercept[18]	6.180	1.010	4.275	8.255	1	8362	6513
Intercept[19]	6.694	1.069	4.714	8.885	1	8480	6670
Intercept[20]	7.441	1.198	5.267	9.931	1	9116	6774
Difference in Hours Played	0.014	0.065	-0.117	0.139	1	5558	4870
Loneliness During Lockdown	0.406	0.102	0.207	0.612	1	8407	6330
Hours by Loneliness	-0.003	0.008	-0.018	0.012	1	5594	4930
Anxiety
Intercept[1]	0.712	0.723	-0.705	2.146	1	10649	6005
Intercept[2]	1.266	0.729	-0.128	2.738	1	11443	6143
Intercept[3]	1.752	0.747	0.340	3.246	1	10733	6512
Intercept[4]	2.138	0.762	0.697	3.649	1	10313	6596
Intercept[5]	2.365	0.776	0.904	3.901	1	9986	6553
Intercept[6]	2.825	0.808	1.319	4.417	1	9425	6504
Intercept[7]	3.055	0.821	1.512	4.674	1	9247	6597
Intercept[8]	3.464	0.841	1.888	5.146	1	9013	6401
Intercept[9]	3.828	0.864	2.216	5.554	1	9010	6355
Intercept[10]	4.026	0.874	2.367	5.749	1	9116	6766
Intercept[11]	4.242	0.882	2.587	6.001	1	9052	6526
Intercept[12]	4.609	0.900	2.916	6.418	1	8953	6605
Intercept[13]	5.082	0.939	3.303	6.945	1	9416	6693
Intercept[14]	5.497	0.975	3.671	7.473	1	9626	6532
Intercept[15]	7.229	1.350	4.836	10.157	1	11167	5835
Difference in Hours Played	-0.024	0.073	-0.171	0.115	1	5228	5204
Loneliness During Lockdown	0.312	0.097	0.126	0.503	1	8942	6273
Hours by Loneliness	0.002	0.008	-0.014	0.020	1	5310	5103
Stress
Intercept[1]	-1.281	0.807	-2.906	0.271	1	6995	6070
Intercept[2]	-0.452	0.743	-1.945	0.983	1	8609	6642
Intercept[3]	-0.023	0.723	-1.470	1.402	1	8587	6431
Intercept[4]	0.420	0.715	-0.995	1.820	1	8235	6887
Intercept[5]	0.611	0.712	-0.784	2.019	1	8351	6816
Intercept[6]	1.027	0.707	-0.347	2.423	1	8324	7044
Intercept[7]	1.251	0.710	-0.132	2.627	1	8185	6848
Intercept[8]	1.612	0.718	0.217	3.035	1	8109	6684
Intercept[9]	1.894	0.725	0.498	3.338	1	8095	6717
Intercept[10]	2.181	0.742	0.743	3.666	1	8229	7112
Intercept[11]	2.469	0.755	1.002	3.976	1	8026	6840
Intercept[12]	2.981	0.777	1.466	4.562	1	7995	6887
Intercept[13]	3.136	0.785	1.614	4.735	1	7960	6821
Intercept[14]	3.564	0.802	2.030	5.214	1	7619	6802
Intercept[15]	3.858	0.821	2.288	5.533	1	7432	6546
Intercept[16]	4.650	0.879	2.931	6.434	1	7397	6530
Intercept[17]	5.032	0.920	3.254	6.889	1	7749	6433
Intercept[18]	5.526	0.976	3.642	7.477	1	8033	6598
Intercept[19]	6.269	1.100	4.205	8.498	1	8661	6772
Difference in Hours Played	-0.004	0.076	-0.157	0.138	1	5265	5555
Loneliness During Lockdown	0.284	0.093	0.104	0.469	1	7989	6554
Hours by Loneliness	0.000	0.009	-0.017	0.017	1	5281	5921

Prior Sensitivity Check

The below plots show how sensitive the parameter estimates and Bayes factors are to different specifications of priors. The models reported in text had the following priors:

	DAS Main Models	Loneliness Main Model	Moderation Models
Intercept	Student-t(3, 0, 2.5)	Student-t(3, 0, 1.5)	Student-t(3, 0, 3)
Beta	Normal(0, 1)	Normal(0, 1)	Normal(0, 1)
SD	Exponential(1)	Exponential(1)

For each model we fitted 10 models in total (including the reported model) with different prior specifications for the Beta (slope) terms. For the main and moderation models these ranged from Normal(0, 0.2) to Normal(0, 2) in steps of 0.2, while for the difference models these ranges from Normal(0, 0.1) to Normal(0, 1) in steps of 0.1.

In the plots below each model is reported showing the sensitivity of both the parameter estimates for each term in the model and for the Bayes factor in support of the point null (i.e. a parameter estimate of 0) relative to the alternative model (i.e. a parameter estimate of \(\neq\) 0). In all cases, the standard deviation for the prior on the beta parameter for the models reported in text is indicated with a vertical dotted line.

Generally, these plots show that the models are not overly sensitive to the prior specification, giving consistent results for the parameter estimates and Bayes factors regardless of whether the beta terms have larger or smaller standard deviations than the models reported.

Study 1

Main Models

There is general agreement between models regardless of the magnitude of the standard deviation for the beta prior for the parameter estimates. While the Bayes factor generally increases as the standard deviation for the beta prior increases, this trend is consistent and conclusions do not change across prior specifications.